Abstract and Applied Analysis
Volume 2006 (2006), Article ID 26724, 43 pages

Robust control problems of vortex dynamics in superconducting films with Ginzburg-Landau complex systems

Aziz Belmiloudi1,2

1Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes1, Rennes Cedex 35042, France
2Center de Maths, Institut National des Sciences, Appliquées (INSA), 20 Avenue des Buttes de Coësmes, CS 14315, Rennes Cedex 35043, France

Received 10 December 2004; Accepted 1 March 2005

Copyright © 2006 Aziz Belmiloudi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We formulate and study robust control problems for a two-dimensional time-dependent Ginzburg-Landau model with Robin boundary conditions on phase-field parameter, which describes the phase transitions taking place in superconductor films with variable thickness. The objective of such study is to control the motion of vortices in the superconductor films by taking into account the influence of noises in data. Firstly, we introduce the perturbation problem of the nonlinear governing coupled system of equations (the deviation from the desired target). The existence and the uniqueness of the solution of the perturbation are proved as well as stability under mild assumptions. Afterwards, the robust control problems are formulated in the case when the control is in the external magnetic field and in the case when the control is in the initial condition of the vector potential. We show the existence of an optimal solution, and we also find necessary conditions for a saddle point optimality.